A flow system is set of paths through which a substance passes, such as from one point to another. Some flow systems, called complex flow systems, can have one or more complexities that affect the flow of the substance. These complexities may include such factors as multiple sources of entry for the substance to enter the system and/or variations that exist between various points (called nodes) in the system. One example of a complex flow system is a river system, which can have such complexities as one or more main channels, which may have nodes with varying cross-sections, slopes, curves, obstructions, and/or the like and/or a set of tributaries that feed the main channel or channels.
Because of the inherent complexities, it can be difficult to track the flow of a substance through a complex flow system. In order to characterize such a complex process, mathematical models, which are defined by a series of equations, input variables and coefficients, can be used. Such models can be developed by calculating flow through a node based on flow from a previous node combined with data regarding the physical features found at the node using a mathematical description of the flow system. Such models often utilize nonlinear partial differential equations based on a set of factors that are meant to represent the features found at each of the nodes.
Unfortunately, most models fail to perfectly forecast the particulars of the flow system in all situations. Rather, some or all of the inputs that are used in these models can contain sources of uncertainty, including, but not limited to: measuring errors, incomplete data, incorrect cause/effect assumptions, etc. Further, the complex flow system itself can change over time (e.g., obstructions, erosion and/or the like, in the case of a river system). Such changes can cause even a model that initially performs relatively accurately to become more and more inaccurate as time elapses. Because of this, mathematical models of complex flow systems often need to be calibrated.